Yes, graphene is an example where the electrons behave as massless particles. In effect, the electrons are described by a relatvistic, massless Dirac equation - i.e a condensed matter version of Q.E.D. To top it off, the corresponding coupling constant is of order 1, hence this QFT is also at strong coupling (like QCD).

Your remark about the high mobility is correct, yes. The electrons are effectively massless, hence an electron cannot be slowed down - only reflected.

I have seen this come up in the forum before, but no there's nothing relativistic about graphene.

To suggest that because of the mysterious "massless" fermions is plain wrong.
Because...
electrons are massless IN the effective mass approximation. The second derivative of E(k) wrt k is 0. And one over that is undefined.

It LOOKS as if the electrons are massless, because of the rather interesting nuclear potential landscape they see around them.

Otherwise they do have mass, and again NO, you don't need Dirac equation per se, to describe them.

High mobility has just to do with the reduced phase space and the fact of low dimensionality.

A carbonnanotube also has out of this world mobilities... But are they relativistic?

NO.

So big reminder: Electrons are massless in the SEMI-CLASSICAL sense, not in the quantum or relativistic sense.

Remember: relativity is about what happens when you change frame of reference, and the ensuing mixing of time and space coordinates. Sound (in air and in solids) are dispersionless too, but we don't go around calling them relativistic.

OK, i might have been a bit sloppy in my remarks. Ofcourse I ment that the picture of massless (Dirac) fermions only arises in an effective model; namely, a thight binding model in which you allow hopping terms between neighbouring sites in a honeycomb lattice.

But this assumption is really all you need. It automatically leads to a picture of Dirac fermions describing the low-lying excitations, i.e. the collective behavior of the electrons (yea, I admit - I wrongly stated that it was the electrons which become relativistic and massless). There is no Lorentz group which leaves the system invariant, but the low-lying excitations are still described by a massless Dirac equation.

I understood the concept I think. In fact, there are so much publications on CNTs but a little say something on the reasons why CNTs are popular and the electrons have high mobility. In fact, also in the book I mentioned at first post (Fundamentals of Nanoelectronics), the term Dirac fermions (that we concluded as wrong) is also said for electrons in CNTs. I now have some suspects about this book :) One must go on by verifying the concepts of the book.

I understood the concept I think. In fact, there are so much publications on CNTs but a little say something on the reasons why CNTs are popular and the electrons have high mobility. In fact, also in the book I mentioned at first post (Fundamentals of Nanoelectronics), the term Dirac fermions (that we concluded as wrong) is also said for electrons in CNTs. I now have some suspects about this book :) One must go on by verifying the concepts of the book.

Cheers,
carbon9

Carbon, great point. I remember I was going crazy to understand why CNTs are so wonderful conductors about a year ago. Then I figured it's a natural consequence of low dimensionality...

Think of this way, suppose you have a bottle of water open in two ends -- allowing steady state water flow through the two ends. Would you want to have the bottle rather fully filled or would you want to have it half filled if you want minimum scattering between water molecules?

In which case there would be ROOM for scattering?... Low dimensionality and natural confinement results in a small density of states. If all the states are already filled in a single moded wire, then there's no room for scattering. Because there are no states to scatter back to.

In a multimoded wire, the possibility is huge, and hence the low mobilities...

CNTs have a Dirac-like dispersion as well as graphene because their dispersion is just 2D slices out of graphene's 3D dispersion. Using the zone folding method, which is just assuming periodic boundary conditions, you essentially just figure out graphene's dispersion, then take slices corresponding to particular allowed k_x values (if the tube direction is k_y).

I prefer, in my publications, to use the phrase 'Dirac-like' dispersion, but a lot of people in the field use the term Dirac Fermions, as have I a few times. The common understanding of course is that there isn't a 1:1 correspondence between low energy graphene / CNTs/ nanoribbons and the relativistic Dirac equation. But as has been pointed out, there are lots of similarities.

The high mobility is particularly significant as well because it remains high at low carrier concentrations.

Anyway, whoever wrote that book, I assume knew what they were talking about when they used the phrase Dirac Fermions!

High mobility in graphene and carbon nanotubes is obviously one of their most useful properties, and it's not simply a matter of low dimensionality. The reason mobility is so high is that electrons (and holes) simply cannot backscatter in the honeycomb lattice. The explanation for this can be thought of in a couple of different ways. If you like to describe low-energy quasiparticle excitations in graphene as being "relativistic," then it is important that they follow the massless Dirac equation, because this makes them chiral. Thus, when you reflect their k-vector, you also have to rotate their spin. As you may recall, a 2pi rotation of a spin-1/2 spinor gives you a minus sign, so the clockwise rotation will destructively interfere with the counter-clockwise rotation, giving you zero backscattering amplitude. This is commonly interpreted as a Berry phase of pi, giving destructive interference of the scattered wavefunction. Refer to Journal of the Physical Society of Japan, Vol. 68, p 2857, 1998, for example.

Or if you find the relativistic description of graphene quasiparticles unsettling, just notice that the forward going and reverse going Blochfunctions in the graphene reciprocal lattice are orthogonal, so it takes something unusual to couple the states (intervalley scattering, broken lattice symmetry, etc). This is not generally true for 2D crystals of different geometry.

With backscattering out of the picture, there is little to impede the movement of electrons in graphene and CNTs, so their mobility is off the charts.

High mobility in graphene and carbon nanotubes is obviously one of their most useful properties, and it's not simply a matter of low dimensionality. The reason mobility is so high is that electrons (and holes) simply cannot backscatter in the honeycomb lattice. The explanation for this can be thought of in a couple of different ways. If you like to describe low-energy quasiparticle excitations in graphene as being "relativistic," then it is important that they follow the massless Dirac equation, because this makes them chiral. Thus, when you reflect their k-vector, you also have to rotate their spin. As you may recall, a 2pi rotation of a spin-1/2 spinor gives you a minus sign, so the clockwise rotation will destructively interfere with the counter-clockwise rotation, giving you zero backscattering amplitude. This is commonly interpreted as a Berry phase of pi, giving destructive interference of the scattered wavefunction. Refer to Journal of the Physical Society of Japan, Vol. 68, p 2857, 1998, for example.

With backscattering out of the picture, there is little to impede the movement of electrons in graphene and CNTs, so their mobility is off the charts.

First of all I don't believe that they (who?) follow the massless Dirac equation. What is the justification of this?

This business (solid state physics) has had nothing relativistic in its entire history. The maximum energy an electron can have in graphene is 10 eV. So what massless Dirac equation are you referring to?

I don't need papers, I just need a clarification on this.

Secondly, look what you are describing... It boils down to REDUCED backscattering, and that is MOST Importantly occurring because of a LACK OF states to back scatter into.

THis is a direct consequence of the LOW dimensionality ( small number of modes )

What makes Graphene interesting is its unique bandstructure. But that just means your Hamiltonian is different, and if you can manage to get a single-moded Silicon nanowire, I claim that the mobility will be no different than graphene.

You can check Landauer formula, ballistic transport, NEGF methodology and other things...

I am pretty sure of what I am saying here, I hope you know what you are talking about when you take graphene to this different category.

And please remember that mobilities as close as to those observed in graphene has been observed also in Silicon nanowires

First of all I don't believe that they (who?) follow the massless Dirac equation. What is the justification of this?

This business (solid state physics) has had nothing relativistic in its entire history. The maximum energy an electron can have in graphene is 10 eV. So what massless Dirac equation are you referring to?

I don't need papers, I just need a clarification on this.

Simple --- they are spin half particles with a linear dispersion. Thus, they follow the Dirac equation at low wavevectors. It's a simple issue of universality.

I agree --- there's nothing relativistic about an electron in graphene. However, there are certain features which usually only appear in high energy electrons, such as the decoupling between different chiralities. I don't ascribe these to relativistic effects, but it's certainly true that relativistic effects also cause them.

We need to move beyond a simple division into "classical" and "relativistic". These are just two possibilities out of a vast collection. Condensed matter is, as always, about the low energy effective behaviour. In this case, at low energies, the excitations are fermionic, have the quantum numbers of electrons, with linear dispersion, and decoupling in the chiral representation. We then appeal to the readers' intuition and say that they are like relativistic particles. This last step is probably wrong, but as a whole, the community understands what is meant, so no corrections are necessary in published articles.

First of all I don't believe that they (who?) follow the massless Dirac equation. What is the justification of this?

This business (solid state physics) has had nothing relativistic in its entire history. The maximum energy an electron can have in graphene is 10 eV. So what massless Dirac equation are you referring to?

You should remember that, for example, Klein tunneling is an effect that could be described as "relativistic". The fact that quasiparticles can overcome arbitrarily high potential barriers in graphene is very easy to reproduce in direct tight-binding calculations, so you don't really have to resort to massless Dirac equation. I'm quite sure that this is not true in single-moded Si nanowires.

It is just the unique bandstructure that reduces down to the Dirac equation in the continuum limit. The electrons in graphene are not relativistic in itself, but the quasiparticles obey an equation of motion similar to that of relativistic electrons in 2D. And therefore people refer to the quasiparticles as relativistic. I do not really see any big problem here.

You should remember that, for example, Klein tunneling is an effect that could be described as "relativistic". The fact that quasiparticles can overcome arbitrarily high potential barriers in graphene is very easy to reproduce in direct tight-binding calculations, so you don't really have to resort to massless Dirac equation. I'm quite sure that this is not true in single-moded Si nanowires.

It is just the unique bandstructure that reduces down to the Dirac equation in the continuum limit. The electrons in graphene are not relativistic in itself, but the quasiparticles obey an equation of motion similar to that of relativistic electrons in 2D. And therefore people refer to the quasiparticles as relativistic. I do not really see any big problem here.

The argument is not really the relativistic context of graphene, or some exotic tunneling effect which is observed under very special circumstances.

Graphene attracts a lot of interest because it has HIGH MOBILITY, therefore in principle it could make a better (faster) transistor than the conventional bulk silicon. Eventhough it has lots of problems, just the slightest possibility is enough to convince people and get FUNDS.

I have drawn the analogy of a nanowire because of a simple fact. Both graphene and silicon nanowires have high mobilities... On the other hand you could say there's spin-orbit coupling in Silicon but not so much of that in graphene (carbon) and that they are really not equivalent (And I don't remember anyone claiming so). So here's another "exotic" difference that nobody really cares.

Graphene and Nanowires are both low-dimensional structures and hence they have high mobility. Yes, you can may be argue that at a 4th order effect of graphene having such and such "special physics" in it helps its mobility.

But the first order and the most important effect is the reduced phase space for scattering. And that's what's the argument is about here.

OK, i might have been a bit sloppy in my remarks. Ofcourse I ment that the picture of massless (Dirac) fermions only arises in an effective model; namely, a thight binding model in which you allow hopping terms between neighbouring sites in a honeycomb lattice.

But this assumption is really all you need. It automatically leads to a picture of Dirac fermions describing the low-lying excitations, i.e. the collective behavior of the electrons (yea, I admit - I wrongly stated that it was the electrons which become relativistic and massless). There is no Lorentz group which leaves the system invariant, but the low-lying excitations are still described by a massless Dirac equation.